However, 10000971767 is prime. And I admit that it wasn't obvious from my earlier post that I was proposing allowing _any_ leading zero to mutate, that is in fact what I had in mind. It doesn't surprise me that there are islands under the rule Edwin Clark (naturally) read me as proposing. I think Rich Schroeppel understood what I meant, because (a) he knows how I think, and (b) he would agree that it's only infinite connectivity that rescues this graph from utter severage. On Thu, Mar 12, 2009 at 6:18 PM, Edwin Clark <eclark@math.usf.edu> wrote:
Even relaxing the adjacency definition to allow 3-->31 and 3-->13, etc, the graph of all primes will not be connected:
If you change any digit of the prime 971767 it becomes composite. Also if d
0 is a digit then all integers of the form d971767 and of the form 971767d are composite.
Edwin
On Tue, 10 Mar 2009, rcs@xmission.com wrote:
3 -> 13:
My notion was that leading 0s are ruled out, so each size is a separate problem space. With this restriction, I think all the two digit primes are connected, and all the three digit primes are connected. The average number of connections is bounded, so there should be a few islands among the larger primes.
The situation is interesting for other radices: Binary just barely hangs together for 5-7, is split for 11,13, and has a stringy graph for 17...29.
Ternary (and other odd radices) split into disconnected pieces, based on the pattern of even and odd digits.
If we allow leading 0s, I'd tend to agree with ACW, but consider one piece of contrary evidence: There's a five-digit number N such that N + 2^K is always composite.
Rich
--------------
Quoting Allan Wechsler <acwacw@gmail.com>:
Do the rules permit you to go from 3 to 13? If yes, then I conjecture
you can get from any prime to any other prime.
On Mon, Mar 9, 2009 at 1:11 AM, Warut Roonguthai <warut822@gmail.com> wrote:
9973 -> 1973 -> 1913 -> 1013 -> 1019 -> 1009
9973 -> 1973 -> 1933 -> 1033 -> 1039 -> 1009 9973 -> 1973 -> 1979 -> 1949 -> 1049 -> 1009 9973 -> 9173 -> 9103 -> 1103 -> 1109 -> 1009 9973 -> 9173 -> 9103 -> 9109 -> 1109 -> 1009
Warut
On Mon, Mar 9, 2009 at 10:30 AM, <rcs@xmission.com> wrote:
The Prime Ladder puzzle is to change one prime into another, by changing a digit at a time. All the intermediate numbers must also be prime.
Example: 101 into 997: 101 -> 107 -> 197 -> 997
Challenge: Change 1009 into 9973.
Rich
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